The book of Linear Algebra that you used is totally OK.

I wish you success in the postgraduate entrance examination! !

20 10 national postgraduate entrance examination mathematics examination outline

Test paper structure (1) Full score of test paper 150, and test time 180 minutes. (2) The proportion of content in higher education is about 80%, and that of linear algebra is about 20%. (3) The proportion of fill-in-the-blank questions and multiple-choice questions is about 40%, and the proportion of solution questions (including proof questions) is about 60%.

Outline of the National Entrance Examination for Mathematics Postgraduates II

[Examination subjects] Advanced mathematics, linear algebra,

Advanced mathematics.

I. Function, Limit and Continuity

Examination content

The Concept and Representation of Function

Boundedness, monotonicity, periodicity and parity of functions

Compound function, inverse function, piecewise function and implicit function.

Properties and graphs of basic elementary functions

Elementary function

The Establishment of Function Relation in Simple Application Problems

Definition and properties of sequence limit and function limit

Left Limit and Right Limit of Function

Concepts of infinitesimal and infinity and their relationship

Properties of infinitesimal and comparison of infinitesimal

Four operations of limit

Two criteria for the existence of limit: monotone bounded criterion and pinch criterion.

Two important limitations

The concept of functional continuity

Types of discontinuous points of functions Continuity of elementary functions

Properties of continuous functions on closed intervals

Examination requirements

1. Understand the concept of function, master the expression of function, and establish the function relationship in simple application problems.

2. Understand the boundedness, monotonicity, periodicity and parity of functions.

3. Understand the concepts of compound function and piecewise function, inverse function and implicit function.

4. Grasp the nature and graphics of basic elementary functions and understand the basic concepts of elementary functions.

5. Understand the concept of limit, the concepts of left limit and right limit of function, and the relationship between the existence of function limit and left and right limit.

6. Master the nature of limit and four algorithms.

7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit.

8. Understand the concepts of infinitesimal and infinity, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.

9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.

10. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.

Second, the differential calculus of unary function

Examination content. The concept of derivative and differential: the relationship between the geometric meaning and physical meaning of derivative and the derivability and continuity of function; Four operations of derivative and differential of tangent and normal basic elementary function of plane curve; differential method of compound function, inverse function, implicit function and function determined by parameter equation; first-order differential invariant differential mean value theorem of higher derivative; monotonicity of extreme value function of L'H?pital's law function; judging concavity and convexity, inflection point and asymptote of function graph; describing maximum and minimum value of function graph; conceptual curvature radius of arc differential curvature.

Examination requirements

1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, find the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and understand the relationship between function derivability and continuity.

2. Master the four algorithms of derivative and the derivative rule of compound function, and master the derivative formula of basic elementary function. Knowing the four algorithms of differential and the invariance of first-order differential form, we can find the differential of function.

3. Understand the concept of higher-order derivative and find the n-order derivative of simple function.

4. Find the first and second derivatives of piecewise function.

5. Find the derivative of implicit function, function determined by parameter equation and inverse function.

6. Understand and apply Rolle theorem, Lagrange mean value theorem and Taylor theorem to understand Cauchy mean value theorem.

7. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, master the method of finding maximum and minimum value of function and its simple application.

8. We can judge the concavity and convexity of the function graph by derivative, find the inflection point and horizontal, vertical and oblique asymptotes of the function graph, and describe the function graph.

9. Master L'H?pital's law to find the limit of indeterminate form.

10. Understand the concepts of curvature and radius of curvature, and calculate curvature and radius of curvature.

3. Integral calculus of unary function

The concept of original function and indefinite integral, the basic properties of indefinite integral, the basic integral formula, the mean value theorem of definite integral, the concept and basic properties of definite integral, the upper limit function of integral and its derivative Newton-Leibniz formula, the substitution integration method of indefinite integral and definite integral, the rational formula of partial integral, the rational function and trigonometric function and the application of integral generalized integral, the definite integral of simple and unreasonable function.

Examination requirements

1. Understand the concepts of original function and indefinite integral and definite integral.

2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the integration methods of method of substitution and integration by parts.

3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.

4. Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula.

5. Knowing the concept of generalized integral, we can calculate generalized integral.

6. Understand the approximate calculation method of definite integral.

7. Grasp some geometric and physical quantities (the area of plane figure, the arc length of plane curve, the volume of rotating body and lateral area, the area of parallel section, known solid volume, work, gravity, pressure) and the average value of functions, and express and calculate with definite integral.

Four, multivariate function calculus

Concept of multivariate function, geometric meaning of binary function, concept of limit and continuity of binary function, concept and calculation of partial derivative of multivariate function in bounded closed region, concepts, basic properties and calculation of extreme value and conditional extreme value of second-order partial derivative of multivariate function, maximum value and minimum value.

Examination requirements

1. Understand the concept of multivariate function and the geometric meaning of bivariate function.

2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.

3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function, total differential, existence theorem of implicit function and partial derivative of multivariate implicit function.

4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Larange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve some simple application problems.

5. Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates). V. Ordinary differential equations

Basic concepts of ordinary differential equations, differential equations with separable variables, homogeneous differential equations, first-order linear differential equations, reducible higher-order differential equations, properties of solutions and structural theorems of solutions, second-order homogeneous linear differential equations with constant coefficients are higher than second-order homogeneous linear differential equations with constant coefficients, and some simple applications of second-order homogeneous linear differential equations with constant coefficients.

Examination requirements

1. Understand differential equations and their concepts such as solutions, orders, general solutions, initial conditions and special solutions.

2. Mastering the solutions of equations with separable variables and first-order linear differential equations can solve homogeneous differential equations.

3. The following equations will be solved by order reduction method: y (n) = f (x), y' = f (x, y') y = f'' (y, y').

4. Understand the properties of the solution of the second-order linear differential equation and the structure theorem of the solution.

5. Master the solution of second-order homogeneous linear differential equations with constant coefficients, and be able to solve some homogeneous linear differential equations with constant coefficients higher than the second order.

6. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order non-homogeneous linear differential equations with constant coefficients.

7. Can use differential equations to solve some simple application problems.

linear algebra

I. Determinants

The concept and basic properties of determinant of examination content; The expansion theorem of determinant by row (column)

Examination requirements

1. Understand the concept of determinant and master its properties.

2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.

Second, the matrix

The concept and properties of the transposed inverse matrix of the determinant matrix of the concept matrix of the test content matrix are the necessary and sufficient conditions for the matrix to be reversible. With the elementary transformation of the matrix matrix, the rank matrix of the elementary matrix is equivalent.

Examination requirements

1. Understand the concepts and properties of matrix, identity matrix, quantization matrix, diagonal matrix, symmetric matrix, triangular matrix and antisymmetric matrix.

2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant of the power of square matrix and the product of square matrix.

3. Understand the concept of inverse matrix, master the properties of inverse matrix, the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find inverse matrix.

4. Understand the concept of elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding matrix rank and inverse matrix by elementary transformation.

Third, the vector

The relationship between the linear combination of concept vectors and the linear correlation between the linear representation vector group and the maximum linearly independent vector group requires the relationship between the rank of the rank vector group and the rank of the matrix.

1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.

2. Understand the concepts of linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.

3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.

4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.

Fourth, linear equations.

Kramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures of Linear Equations; Basic solution system of homogeneous linear equations and general solution of nonhomogeneous linear equations.

Examination requirements

The length can be determined by Cramer's law.

2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.

3. Understand the concepts of basic solution system, general solution and solution space of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations.

4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.

5. We can use elementary line transformation to solve linear equations.

Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix

Concept and property similarity transformation of eigenvalues and eigenvectors of test content matrix, necessary and sufficient conditions for similar diagonalization of concept and property matrix, eigenvalue, eigenvector and similar diagonal matrix of real symmetric matrix of similar diagonal matrix.

Examination requirements

1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix.

2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions for similar diagonalization of matrix, and transform the matrix into similar diagonal matrix.

3. Understand the properties of eigenvalues and eigenvectors of real symmetric matrices.